Hyper-Mahler measures via Goncharov-Deligne cyclotomy

Abstract

The hyper-Mahler measures $m_k(1+x_1+x_2),k\in {\mathbb{Z}}_{>1}$ and $m_k(1+x_1+x_2+x_3),k\in {\mathbb{Z}}_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb{Q}$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $3$, $4$, $6$, $8$, $10$ and $12$.

(v1) i+30 pages, 5 tables. (v2) i+37 pages, 7 tables. Results improved and enriched. Maple and Mathematica worksheets available as ancillary files. (v3) 47 pages, 8 tables. Reformatted and corrected. (v4) 51 pages, 8 tables. Accepted version

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